jmc

Question

In the adjoining figure, two circles with radii 8 and 6 are drawn with their centers 12 units apart. At P, one of the points of intersection, a line is drawn in such a way that the chords QP and PR have equal length. Find the square of the length of QP. FIGURE

Accepted Answer

Let QP=PR=x. Angles QPA, APB, and BPR must add up to 180^. By the Law of Cosines, APB=^-1(-1124). Also, angles QPA and BPR equal ^-1(x16) and ^-1(x12). So we have ^-1(x16)+^-1(-1124)=180^-^-1(x12). Taking the cosine of both sides, and simplifying using the addition formula for as well as the identity ^2x + ^2x = 1, gives x^2=130.